Tuanaki, Kaupae 3, 2017 · cosecc osec cot cotc osec () d 1 (1 ) 1 ln () ln d d d d.d d d d d d d d.d d ax ax n n 2 2 1 2 2 ALGEBR Qu adr tics ... Cone Volume = 1 3!r2h Curved surface
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sin(A ± B) = sin Acos B ± cos Asin Bcos( A ± B) = cos Acos B sin Asin B
tan( A ± B) =tan A ± tan B
1 tan A tan B
sin2A = 2sin Acos A
tan 2A =2 tan A
1 tan2 A
cos2A = cos2 A sin2 A
= 2cos2 A 1
= 1 2sin2 A
TE PĀKOKI
Te Ture Aho
Te Ture Whenu
Ngā Whārite ka Pono Ahakoa ngā Uara Ka Whakaurua Atu
Ngā Otinga Whānui
Ngā Koki Hiato
Ngā Koki Rearua
±
±
cos2 θ + sin2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ
Mēnā sin θ = sin α kāti θ = n� + (−1)n αMēnā cos θ = cos α kāti θ = 2n� ± αMēnā tan θ = tan α kāti θ = n� + αko te n, he tau tōpū ahakoa
Ngā Otinga Whakarau
22
sin cos sin( ) sin( )cos sin sin(
A B A B A BA B A
= + + −= + BB A B
A B A B A B) sin( )
cos cos cos( ) cos( )sin
− −= + + −2
2 AA B A B A Bsin cos( ) cos( )= − − +
Ngā Otinga Tāpiri
sin sin sin cos
sin sin cos
C D C D C D
C D C D
+ = + −
− = +
22 2
22
ssin
cos cos cos cos
cos cos
C D
C D C D C D
C D
−
+ = + −
− =
2
22 2
−− + −22 2
sin sinC D C D
TE INE
Te Tapatoru
Te Taparara
Te Pewanga
Te Rango
Te Koeko
Te Poi
Horahanga =
Horahanga =
Horahanga =
Te roa o te pewa = rθ
Rōrahi = πr 2hHorahanga mata kōpiko = 2πrh
Rōrahi =
Rōrahi =
r 2θ
Horahanga mata kōpiko = πrl ina ko te l te teitei o te tītaha
πr 2h
Horahanga mata = 4πr 2
πr 343
13
12
12
12
(a + b)h
absinC
cosec θ =1
sinθ
sec θ =1
cosθ
cot θ =1
tanθ
cot θ =cosθsinθ
asin A
=b
sin B=
csinC
c2 = a2 + b2 2ab cosC
sin(A ± B) = sin Acos B ± cos Asin Bcos( A ± B) = cos Acos B sin Asin B
tan( A ± B) =tan A ± tan B
1 tan A tan B
sin2A = 2sin Acos A
tan 2A =2 tan A
1 tan2 A
cos2A = cos2 A sin2 A
= 2cos2 A 1
= 1 2sin2 A
TE PĀKOKI
Te Ture Aho
Te Ture Whenu
Ngā Whārite ka Pono Ahakoa ngā Uara Ka Whakaurua Atu
Ngā Otinga Whānui
Ngā Koki Hiato
Ngā Koki Rearua
±
±
cos2 θ + sin2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ
Mēnā sin θ = sin α kāti θ = n� + (−1)n αMēnā cos θ = cos α kāti θ = 2n� ± αMēnā tan θ = tan α kāti θ = n� + αko te n, he tau tōpū ahakoa
6
1
1
1
2 3
2
� 6 –
� 3 –
� 4 –
� 4 –
TRIGONOMETRY
cosec! = 1sin!
sec! = 1cos!
cot! = 1tan!
cot! = cos!sin!
Sine Rulea
sin A= b
sinB= c
sinC
Cosine Rule
c2 = a2 + b2 ! 2ab cosC
Identities
cos2! + sin2! = 1
tan2! +1= sec2!
cot2! +1= cosec2!
General Solutions
If sin ! = sin " then ! = n! + ("1)n" If cos ! = cos " then ! = 2n! ±"If tan ! = tan " then ! = n! +"where n is any integer
Compound Anglessin(A± B) = sin AcosB ± cos AsinBcos(A± B) = cos AcosB ! sin AsinB
tan(A± B) = tan A± tanB1! tan A tanB
Double Anglessin2A = 2sin Acos A
tan2A = 2 tan A1! tan2 A
cos2A = cos2A! sin2A
= 2cos2A!1
= 1! 2sin2A
Products2sin Acos B = sin( A+ B) + sin( A − B)2cos Asin B = sin( A+ B) − sin( A − B)2cos Acos B = cos( A+ B) + cos( A − B)2sin Asin B = cos( A − B) − cos( A+ B)
Sums
sinC + sin D = 2sinC + D
2cos
C − D2
sinC − sin D = 2cosC + D
2sin
C − D2
cosC + cos D = 2cosC + D
2cos
C − D2
cosC − cos D = −2sinC + D
2sin
C − D2
MEASUREMENTTriangle
Area =12
absinC
Trapezium
Area =12
(a + b)h
Sector
Area =12
r2θ
Arc length = rθ
Cylinder
Volume = !r2hCurved surface area = 2!rh
Cone
Volume =13!r2h
Curved surface area = !rl where l = slant height
Sphere
Volume =43!r3
Surface area = 4!r2
7
Level 3 Calculus, 2017
9.30 a.m. Thursday 23 November 2017
FORMULAE AND TABLES BOOKLETfor 91577, 91578 and 91579
Refer to this booklet to answer the questions in your Question and Answer booklets.
Check that this booklet has pages 2 – 7 in the correct order and that none of these pages is blank.
YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION.
L3
–C
AL
CM
F
English translation of the wording on the front cover